Unit Circle
In short: The unit circle has radius 1 and sits on the origin; rotating to an angle θ lands on the point (cos θ, sin θ), so sines and cosines become plain coordinates.
Algebra 2TrigonometryF-TF.A.2
What is Unit Circle?
The unit circle is a ruler for angles. Begin at (1, 0), walk counterclockwise around the rim by an angle \(\theta\), and read your landing spot: its horizontal coordinate is \(\cos\theta\), its vertical coordinate is \(\sin\theta\).
The curve obeys \(x^2 + y^2 = 1\). Turning counterclockwise from the positive x-axis counts as a positive angle, whether you label it in degrees or radians. With a radius of exactly 1, the ratio \(\tan\theta = \frac{\sin\theta}{\cos\theta}\) is just the slope of the radius.
How to spot it
- A question asks for exact values of sine or cosine at angles like 30°, 45°, 60° or their multiples.
- An angle is given in radians, such as 4π/3, and a point on a circle is wanted.
- A problem lists coordinates like (-1/2, -√3/2) and asks which angle they belong to.
- You need the sign of a trig value and must decide which quadrant the angle ends in.
The method
- Convert the angle to degrees if helpful and locate its terminal side by quadrant.
- Find the reference angle between the terminal side and the x-axis.
- Use the special-angle values \(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\), \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\), \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) for the first quadrant.
- Attach signs from the quadrant: cosine is the x-sign, sine is the y-sign.
Unit Circle: solved example
Problem
Find the coordinates of the point on the unit circle at \(\theta = \frac{4\pi}{3}\) radians, and verify that it lies on the circle.
Solution
Because \(\frac{4\pi}{3}\) radians equals \(240^{\circ}\), the ray lands 60° beyond the negative x-axis. With a 60° gap, the magnitudes are \(\frac{1}{2}\) for cosine and \(\frac{\sqrt{3}}{2}\) for sine, and Quadrant III makes both negative.
The point is \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\). Check: \(\frac{1}{4} + \frac{3}{4} = 1\).
Try it
Question 1
Find \(\tan\frac{3\pi}{4}\) using the circle.
Show the answer
The point is \(\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\), so \(\tan = \frac{\sqrt{2}/2}{-\sqrt{2}/2} = -1\).
Question 2
A point \(\left(\frac{3}{5}, y\right)\) lies on the circle in Quadrant IV. Find \(y\).
Show the answer
\(y^2 = 1 – \frac{9}{25} = \frac{16}{25}\), and in Quadrant IV \(y\) is negative: \(y = -\frac{4}{5}\).
Unit Circle vs. Radian
The circle is a drawing: every point exactly 1 away from the origin. A radian is a size of turn, namely the turn that sweeps out a curved distance of 1 along it. Radians label locations here, yet a measuring unit and a shape are different things.
Keep going
Lessons
Worksheets and quizzes
- Radians and the Unit Circle – Worksheet
- Radians and the Unit Circle – Quiz
- Radians and the Unit Circle – Worksheet
- Radians and the Unit Circle – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Trigonometry